EDIT (Harry): Since this question in its original form was poorly stated (asked about topology rather than graph theory), but we have a list of Topology books in the answers, I guess you should go ahead and post with regard to that topic, rather than graph theory, which the questioner can ask again in another topic.

EDIT (David): The original question was asking for places to learn topology with an eye towards applying it to computer science (artificial neural networks in particular)

What does this question have to do with algebraic topology? Your question is very vague.
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Gjergji ZaimiDec 10 '09 at 12:03

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My impression is that the word "topology" means something different in network theory than it does in mathematics. It seems to refer to the large-scale properties of a graph.
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Qiaochu YuanDec 10 '09 at 15:19

The answers are addressing the wrong question, though. I think that what the OP wants to know about is Graph Theory.
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José Figueroa-O'FarrillDec 10 '09 at 15:48

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I'm pretty sure the OP wants to know neither about topology nor graph theory. When someone says "The key element of ANN is topology" they don't mean (the theory of) topology, or graph theory. They mean that an important component of designing neural networks is choosing the connectivity of the neurons. The best place to learn about this is probably the neural network literature. Having said that, there's some nice interplay between neural networks and topology in the area of detecting topological invariants of images. (Eg. it's easier to learn to detect Euler number than connectivity.)
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Dan PiponiDec 10 '09 at 19:42

15 Answers
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A self study course I can recommend for topology is Topology by JR Munkres followed by Algebraic Topology by A Hatcher (freely and legally available online, courtesy of the author!). But that is if you want to be able to really do the math in all its glorious detail. Basic Topology by MA Armstrong is a shortcut and a very good one at that.

The closest I can get to what you are asking for here is Network Topology. Is that what you mean? In that case you should be probably be looking at topological graph theory. Wikipedia also tells me that something called Computational Topology exists, but that is probably not what you are looking for.

Since the discussion has broadened from the original question to include a wider range of topology books, let me add one more. This is an algebraic topology book by Tammo tom Dieck published just a year ago with the canonical title "Algebraic Topology". Its viewpoint is fairly homotopy-theoretic, as in May's book, and it has a similar density coefficient that some commenters here seem to like. What really impressed me about the book is that in the last few chapters the author manages to give the first ever non-spectral-sequence proofs of some deep and fundamental theorems like Serre's theorem that the homotopy groups of spheres are finitely generated, and Serre's calculation of all the non-torsion. Another is the Hirzebruch signature theorem, the very last theorem in the book. These results are 50 years old, yet apparently no one had previously seen how to prove them without spectral sequences. Of course, spectral sequences are important things that serious topologists should know about, and their use cannot always be avoided, but it's illuminating to see when they are needed and when they are not. Whenever I get around to a second edition of my book I'll have to include tom Dieck's new approach, and I think one can go even further and develop the basic framework of rational homotopy theory without spectral sequences.

It's too bad that math books aren't like Google Maps where one can zoom in or out to get the level of detail and density one wants, or switch between satellite and map views to include or omit things like examples and informal discussions of ideas and motivation. Maybe someday this sort of thing will be possible with electronic books.

I learned general topology with a book that only contained theorems and I referred to Kelley when I had too much difficulty proving them. This method of learning works very well with General topology since most of the proofs are straight forward.

For general topology, one of my favorites is James Dugundji´s General Topology from 1966. It covers pretty much every topic from general topology, it is thorough, dry and has well optimized proofs (the shortest and 'nicest'), no distracting "bla-bla".

The best introduction I know to the entire field of topology is John McCleary's A First Course in Topology: Continuity and Dimension.Not only does it present all the essentials in a strongly geometric manner in low dimensions,it gives a historical perspective on the subject.

What's the best introduction to algebraic topology?

Well,depends on if you like geometric intuition or not. If so,Allen Hatcher's textbook is considered by many to be the new gold standard. And best of all,it's available online for free at Hatcher's website.

If you like more modern (i.e. abstract) approaches,the book by Joseph Rotman can't be beat. And recently,an awesome text by Tammo tom Dieck came out which is probably the state of the art right now and is very readable.

A book that's probably too difficult to use as a textbook but is so beautiful that it needs to be used as a supplement is Peter May's A Concise Course In Algebraic Topology.By May's own admission,it's probably too tough for a first course on the subject,but it is beautifully written and gives a great overview of the subject. It also has a very good bibliography for further study.

My favorite texts on algebraic topology?Probably the 2 books by V.V. Pravalov, Elements of Combinatorial And Differential Topology and Elements of Homology Theory.both available in hardcover from the AMS.Together,they probably give the single most complete presentation of topology that currently exists,with plenty of low-dimensional pictures,concrete constructions and emphasis on manifolds.

And of course,I'd be remiss if I didn't mention the wamdering oddball text which a lot of US universities are afraid to use for thier first course,but is a treasure trove for mathematics students: John Stillwell's Classical Topology And Combinatorial Group Theory. An incredibly rich historical presentation by a master. It's strange organization and selection of material is a double edged sword-but it will give amazing insight to the basic ideas of topology and how they develped. These tools will give a student coming out of Stillwell's book a very strong foundation for studying more modern presentations. I highly recommend it to anyone interested in topology at any level.

There are also several terrific free online lecture sources you should look at,primarily the complete notes of K.Wurthmuller and Gregory Naber. Both can be found at Math Online and I recommend them both highly.

For a very accessible introduction to topology, I recommend Sutherland: "Introduction to Metric and Topological Spaces" (amazon UK). It's incredibly well written and packed with examples and exercises.

I used that book for a class, and I really dislike it. It feels like slightly beefed up analysis book. There's too much focus on metric spaces, and basically nothing on separation axioms, etc.
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Harry GindiDec 11 '09 at 10:58

Two books to add to the already comprehensive list. However, these books focus less on building a rigorous treatment of topology and more on building intuition.

The first, for geometric topology, is A Topological Aperitif. I read this book a while back and really enjoyed the geometric flavor, and it introduced some useful concepts for topology.

The second, for algebraic topology, is An Intuitive Approach. This books covers almost all of the major topics of algebraic topology, using very intuitive explanations in about 140 pages. Most proofs are left out, but intuition for this material is very nice. He covers all the basics except covering spaces and higher homotopy groups.

For general topology, I've always liked the first part of Bredon's Topology and Geometry. Munkres's book is beautifully written but I dislike its architecture. It lingers for too long and outstays its welcome. In contrast, Bredon's book develops the basic material at a rapid rate without skimping on necessary detail while covering important technical results like Baire's category theorem and Urysohn's lemma.

In my opinion, for general topology there is no better text than Runde's "A Taste of Topology" since it is short and comprehensive enough to learn it all (in contrast to Munkres's which is a good book but I think too tedious and long to master quickly). Then for a faster review and a good introductory but thorough reference on algebraic topology I would pick Bredon's "Topology and Geometry" without any doubt. A nice short clarifying companion is Sato/Hudson's "Algebraic Topology: an intuitive approach".
Beyond that, the books by tom Dieck or May as recommended in other answers.

tom Dieck's book would make a better modern course then May since it's slower and more comprehensive. But unfortunately,its prerequisites are considerably higher.A very strong 2 year graduate topology sequence,to me,would begin with a thorough coverage of John Lee's INTRODUCTION TO TOPOLOGICAL MANIFOLDS before moving on to tom Dieck or Rotman. I have mixed feelings about Bredon's book. I'm really impressed with the attempt to mix a purely abstract,modern approach with geometric content on manifolds,but I don't think it's entirely successful. I think Bott/Tu does a much better job of this.
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Andrew LJul 14 '11 at 19:07

Edit: Some people have suggested that you might want a general topology book. If so, read Bourbaki's book on topology. It's probably the most modern treatment of the subect even though it was published some 50-60 years ago. It uses filters to work with more general notions of convergence, and defines metric spaces in terms of uniform spaces, which is where they should naturally live. It's very systematic and very dry, but it's an absolutely excellent book.

I'm only going to say this very vaguely, and don't ask me where, but I've HEARD from people that someone recently scanned both books and that they're floating around on the internet. I can't say anything else.

I have to strongly disagree with the recommendation of Bourbaki. While parts of Bourbaki have stood the test of time (the volume dealing with Coxeter groups and root systems is still a great source), the volume on general topology has not. It's not "modern" so much as idiosyncratic. For instance, metric spaces are important everywhere in math, while uniform spaces are (almost) a dead end. If you want to learn point-set topology, then you should read Munkres or Kelly (which also dicusses convergence in terms of filters and nets) or Dugundji. In my experience, most students prefer Munkres.
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Andy PutmanDec 10 '09 at 20:27

Filters are like nets but better, in my opinion. Also, just because uniform spaces never went anywhere doesn't mean that it's not a good way to deal with spaces that aren't metrisable.
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Harry GindiDec 10 '09 at 22:28

Well,having attended Melvyn Nathanson's seminar where he gave Glaser's proof of Hindman's theorum using ultrafilters,you won't get any arguement from me on thier utility,Harry. I just think nets are easier for beginners since they're such a simple generalization of sequences.
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Andrew LApr 11 '10 at 19:04

And May's book is way too hard for most beginners and leaves too many topics omitted.
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Andrew LJul 31 '10 at 5:28

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@Andy: If uniform spaces are a dead end, then how do you complete a non-metrizable topological vector space?
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Dmitri PavlovSep 11 '11 at 18:42

This question originally asked about places to learn topology with an eye towards applying it to programming. The OP then clarified that he was interested in Artificial Neural Networks. Most answers have been about which textbooks are good for topology, but none address connections to computer science. One mentions computational topology, which seeks to develop efficient algorithms for solving topological problems, e.g. computing homology and homotopy. This doesn't seem to be what the OP was after, since he wanted to apply topology to computer science and not the other way around.

As an algebraic topology PhD student who's simultaneously getting a masters in CS, I have thought quite a bit about ways to combine the two. I've seen talks in which algorithms are developed to apply homology to detect information about graph properties, but these are mostly useless in practice because they're so slow. I haven't seen any algorithms using more than $H_1$ except in the work of Robert Ghrist, who uses sheaf cohomology and similarly high-brow mathematics to do pretty impressive things with sensor networks and engineering (e.g. flows). He seems to be the primary pioneer in this new field and calls it Applied Topology. He recently updated his website to include lots of powerpoints and pdfs to help a beginner get into this field. Here is the link. I've also seen some conferences about Applied Topology in the works, so keep your eyes open for them if you're interested in this field.